Bootstrapping Unstable First-Order Autoregressive Processes

Bootstrapping continuous-time autoregressive processes

We develop a bootstrap procedure for Lévy-driven continuous-time autoregressive (CAR) processes observed at discrete regularly-spaced times. It is well known that a regularly sampled stationary Ornstein–Uhlenbeck process [i.e. a CAR(1) process] has a discrete-time autoregressive representation with i.i.d. noise. Based on this representation a simple bootstrap procedure can be found. Since regul...

Autoregressive Gaussian Random Vectors of First Order

Bootstrapping General First Order Autoregression

In this paper we consider general rst order autoregression, including the stationary, the explosive and the unstable case. It is well-known in the literature that the usual bootstrap method for the least squares parameter estimator is asymptotically consistent for the stationary and the explosive case, but does not work in the unstable case, where the parameter value is equal to + 1 or {1. We p...

Unstable First-order Theories

In this paper, we examine the divide between stable and unstable first-order theories in model theory. We begin by defining a stable theory and proving Shelah’s theorem, which reduces the question of stability to a problem of examining a single formula. Afterwards, we will provide some applications of the stability/instability divide to other model-theoretic questions, such as the question of c...

autoregressive gaussian random vectors of first order